UBC Mathematics 256 Examination - April 2010 - Differential Equations, Exams of Differential Equations

A cover page for a closed-book examination in differential equations at the university of british columbia (ubc) from april 2010. The examination includes instructions, rules, and 5 problem sets covering topics such as fourier series, boundary value problems, phase plane analysis, and laplace transforms.

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The University of British Columbia
Sessional Examinations - April 2010
Mathematics 256
Differential Equations
Closed book examination Time: 21
2hours
Name Signature
Student Number Instructor’s Name
Section Number
Special Instructions:
One 81
2×11” two sided cheat sheets are permitted.
One 81
2×11” one sided table of Laplace transforms is permitted.
Non-programmable calculators allowed.
Show your work in the spaces provided.
You are encouraged to explain all steps in your solutions.
Rules Governing Formal Examinations
1. Each candidate must be prepared to produce, upon request, a library/AMS card
for identification.
2. Candidates are not permitted to ask questions of the invigilators, except in cases
of supposed errors or ambiguities in examination questions.
3. No candidate shall be permitted to enter the examination room after the expiration
of one half hour from the scheduled starting time, or to leave during the first half hour
of the examination.
4. Candidates suspected of any of the following, or similar, dishonest practices shall
be immediately dismissed from the examination and shall be liable to disciplinary
action.
(a) Having at the place of writing any books, papers or memoranda, calculators,
computers, audio or video cassette players or other memory aid devices, other than
those authorized by the examiners.
(b) Speaking or communicating with other candidates.
(c) Purposely exposing written papers to the view of other candidates. The plea of
accident or forgetfulness shall not be received.
5. Candidates must not destroy or mutilate any examination material; must hand
in all examination papers; and must not take any examination material from the
examination room without permission of the invigilator.
1 20
2 20
3 20
4 20
5 20
Total 100
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Be sure that this examination has 11 pages including this cover

The University of British Columbia Sessional Examinations - April 2010

Mathematics 256 Differential Equations

Closed book examination Time: 2 12 hours

Name Signature

Student Number Instructor’s Name

Section Number

Special Instructions:

One 8 12 ” × 11” two sided cheat sheets are permitted. One 8 12 ” × 11” one sided table of Laplace transforms is permitted. Non-programmable calculators allowed. Show your work in the spaces provided. You are encouraged to explain all steps in your solutions.

Rules Governing Formal Examinations

  1. Each candidate must be prepared to produce, upon request, a library/AMS card for identification.
  2. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions.
  3. No candidate shall be permitted to enter the examination room after the expiration of one half hour from the scheduled starting time, or to leave during the first half hour of the examination.
  4. Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and shall be liable to disciplinary action. (a) Having at the place of writing any books, papers or memoranda, calculators, computers, audio or video cassette players or other memory aid devices, other than those authorized by the examiners. (b) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness shall not be received.
  5. Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take any examination material from the examination room without permission of the invigilator.

Total 100

Marks

[20] 1. The function f (x) is defined for 0 ≤ x ≤ 2 by f (x) = x(2 − x). (a) Sketch the odd and even extensions to f (x) over the range − 2 ≤ x ≤ 2. Would you expect the Fourier sine series to converge to f (x) faster or slower than the Fourier cosine series? Explain your reasoning in 1-2 sentences.

(b) Compute the Fourier sine series to f (x).

Marks

[20] 2. (a) Find the general solution of the following homogeneous linear system:

x′^ =

x,

and sketch the phase plane close to x = 0. Is this point a saddle, a node, a spiral or a centre?

(b) Find the solution of the following initial value problem:

x′^ =

x +

−e−^2 t − 6 e−^2 t

, x(0) =

What is unusual about the initial conditions?

(c) Express the steady state response from part (b) in phase-amplitude form: yp(t) = R cos(ωt − φ 0 ), showing that the amplitude R depends only on ω^2. What value of ω within the range 1 ≤ ω ≤ 5 will maximize the amplitude?

[20] 4. Consider the the following initial boundary value problem

ut = uxx − 2 , 0 < x < 2 , t ≥ 0 ,

subject to the boundary and initial conditions:

u(0, t) = 1, u(2, t) = 1, u(x, 0) = 2

(a) Writing u(x, t) = us(x) + v(x, t) state the problems that are satisfied by the steady state solution, us(x), and the transient solution v(x, t).

(b) Find the steady state solution, us(x).

[20] 5. (a) Find the Laplace transform of f (t) = t sin t.

(b) Find the inverse Laplace transform of

F (s) =

5 s + 10 s^2 + s − 6

Solve the following initial value problems using Laplace transforms and describe the behaviour of y(t) as t → ∞:

(c) y′′^ + 8y′^ + 12y = 0, y(0) = 1, y′(0) = 0

.

(d) y′′^ + 8y′^ + 12y = δ(t − 2) + 2u 1 (t) − u 3 (t), y(0) = 0, y′(0) = 0

.

The End